PL174299B1 - Power converter for providing dc power supply to an arc furnace - Google Patents

Abstract

This device includes at least one transformer (9, 10, 11) supplied, at its primary (9) by a three-phase alternating current and delivering, on at least one secondary (10, 11), a three-phase current applied to rectifying means delivering, as output to the load, a rectified voltage and current, these rectifying means being of the type including, for each secondary, controlled semiconductor devices (15, 16). According to the invention, the rectifying means comprise a free-running circuit (19), the said controlled semiconductor devices being triggered with firing angles which are essentially variable, these being modified so as to increase the conduction period in the free-running circuit while at the same time decreasing the conduction period in the triggered semiconductor devices, and conversely, thus making it possible to deliver, to the load, an active power or a reactive power which is substantially constant despite variations in the impedance of the load. <IMAGE>

Description

The present invention relates to a power converter for the DC supply of an electric arc furnace.

Electric arc furnaces are mainly used for the melting and refining of scrap.

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The solution according to the invention is used especially in the direct current supply of furnaces whose electrical parameters are particularly susceptible to frequent and considerable fluctuations.

A method of supplying electricity to an arc furnace that has long been known is to directly supply the electrodes with a three-phase alternating current through a first step-down transformer which feeds a system of busbars distributing the current to at least one adjustable voltage-down transformer and directly energizes the furnace. The furnace has a scrap container inside which three movable electrodes are placed, each connected to one of the phases of the secondary winding of the second step-down transformer. The resulting arc between the electrodes and the scrap metal melts and refines the scrap.

Such an AC-powered furnace draws a substantially constant power, but the fluctuation in reactive power and amperage is considerable. Work takes place at substantially constant active power, except short circuit operation. In particular, significant fluctuations in reactive power can cause large fluctuations in the voltage of the supply network, causing electrical disturbances in it.

There is a tendency to replace the AC power supply with DC power, using an AC to DC converter, between the second step-down transformer and the electrodes.

DC power supply has two major advantages. One is that the positive pole of the arc is formed by the scrap itself, in solid or molten form, and the moving electrode, one or more, forms the negative pole. The energy dispersion along the arc is non-linear and is greater on the positive pole side, thus heating the scrap more than the electrode. A second advantage is that the occurrence of short-circuit overcurrents, very frequent in the switch-on and melting steps, is limited by the transducer. This entails less wear on the electrodes, a reduction in electrical noise introduced into the grid, especially flicker, and, consequently, an improvement in the efficiency of the furnace.

A DC converter used hitherto is known, for example, from the patent description FR-2,588,432. In a converter of this type, the primary winding of the transformer feeds at least one secondary winding, which is connected to a rectifier bridge, corresponding to a typical Graetz bridge, with classical controlled semiconductor devices, preferably thyristors. One of the output terminals of each of the bridges is connected to a common moving electrode and the other terminal is connected, via an equalizing coil, to the bottom electrode of the furnace and is in direct contact with the scrap. In order to limit the harmonics introduced into the network, numerous Graetz bridges for six pulsations are used, each of which is powered with a phase shift of the transformer secondary windings.

This known device operates at constant current with large fluctuations in active power, unlike an alternating current supply. It can be seen reduced, but still significant, fluctuations in reactive power. Moreover, in average values, the consumption of reactive energy is increased, which as a rule dictates the presence of a regulator for the second step-down transformer and a compensation unit. Such a DC device does not offer any significant improvement over an AC device.

The power converter according to the invention is used for the direct current supply of an electric arc furnace. It is provided with at least one transformer fed from the side of the three-phase primary windings with alternating current, which has at least one secondary winding through which the three-phase current flows and which is connected to a rectifier unit, the direct current output of which is connected to the furnace charge. The rectifier unit, associated with each transformer secondary winding, is equipped with controllable semiconductor devices. A converter of this type is characterized in that an energy flow control circuit of the flywheel type is connected between the midpoint of the transformer secondary winding and the DC output terminals of the rectifier unit, and the controlled semiconductor devices are switched on at different switching angles to increase and decrease the instrument conduction time. on state semiconductor.

It is preferred that the energy flow control circuit is a diode circuit, and the transducer is irreversible.

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Rectifier assemblies connected to the transformer secondary windings are connected to each other in an offset arrangement with different switching angles of the controlled semiconductor devices.

The rectifier unit is preferably provided with a sub-assembly for determining the switching-on angles of the controlled semiconductor devices adapted to minimize the reactive power consumed at a minimum average value per cycle, for a given operating point, possibly to minimize the reactive power fluctuations around its mean value, for a given rectified current, possibly to be kept at constant level of the average reactive power consumption in the event of fluctuations in rectified current.

It is preferred that the rectifier units are connected to each other in a parallel offset circuit. Moreover, the rectifier assemblies are adapted to cyclically change the shift direction.

It is preferable that the transducer is additionally provided with current regulators of the two rectifier units, the magnitudes of these currents being substantially equal.

It is preferable that the converter is provided with a set of rectifier devices, the associated transformers of which are energized in phase shift.

The DC converter according to the invention makes it possible to significantly reduce the reactive power consumption and significantly improve the current-voltage characteristics of the DC arc, while reducing the total cost of the converter, by simplifying the transformer directly feeding the furnace, since the regulator is no longer necessary and the dimensions of the compensation unit, usually by half.

The DC converter according to the invention allows operation with an essentially constant active power, such as in an AC furnace, but with a low reactive power consumption and without the excessive expansion of the converter and its associated transformer.

According to the invention, the reactive power - active power characteristic can be influenced in a simple manner in order to optimize it, depending on the method of use of the furnace.

It is possible, for example, to obtain a characteristic that minimizes the value of reactive power absorbed for a given operating point, or a characteristic that minimizes fluctuations in reactive power around its average value, in particular in the case of supply networks that are too small, for which reduction of flicker in the network is important. task, while increasing efficiency and reducing installation costs.

Furthermore, the operating characteristics of the furnace can be selected such that the reactive power absorbed remains substantially constant despite possible current fluctuations, in particular in the event of fluctuations in the charge impedance.

The subject of the invention is described in detail in the exemplary embodiments in the drawing, in which Fig. 1 shows schematically a known configuration of an arc furnace powered directly by alternating current, Figs. 2 and 3 show, respectively, the voltage-current characteristic and the characteristic reactive power - active power of a known electric-powered furnace. 4 shows a schematic configuration of an arc furnace powered by a direct current from a known converter, fig. 5 - the known Graetz bridge circuit in more detail, figs. 6 and 7 show the voltage-current characteristic and the reactive power-active power characteristic curve, respectively 4, Fig. 8 shows a schematic configuration of a DC arc furnace with a converter according to the invention, Figs. 9 and 10 show known configurations of offset coupling circuits of a plurality of transducers of the same structure, Figs. 12 and 13 show the voltage characteristics respectively - current and characteristics reactive power - active power of a furnace supplied with direct current by a converter according to the invention, having the possibility of changing the angles of Graetz bridges switching on, which angles are shown in the lower part of Fig. 13, Fig. 14 shows various characteristics reactive power - active power, which are obtainable by selecting the appropriate switching angles to adapt the operation of the furnace to different types of use, and Fig. 15 shows a family of characteristics reactive power - active power obtained for different values of rectified current, resulting from the reduction of the charge impedance.

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As shown in Fig. 1, a known furnace configuration is to directly supply the electrodes with three-phase alternating current through a first step-down transformer 1 and a busbar system 2, which distribute the current to at least one regulated step-down transformer 3 and directly supply the furnace 4. The furnace 4 consists of a tank 5 containing in its inner part scrap 6, inside which are placed three mobile electrodes 7, each of which is connected to one of the phases of the secondary winding of the regulated transformer 3, causing an arc 8 between electrodes 7 and the scrap. 6, which in turn melts and refines the scrap.

Figures 2 and 3 show the operating characteristics of such an AC electric arc furnace, the dependence of the voltage U on the current I and the dependence of the reactive power Q on the active power P, for different operating points. The CC point represents the short circuit operating point, F - the range corresponding to the melting of the scrap, A - the range corresponding to the refining of the scrap.

As can be seen, such an alternating current furnace draws a substantially constant power, i.e. AP is small, but the fluctuation of the reactive power AQ and the current intensity ΔΙ are considerable. Work takes place at substantially constant active power, except short circuit operation. In particular, significant fluctuations in reactive power can cause significant fluctuations in the voltage of the supply network, causing electrical disturbances in it, in particular in the form of flicker, i.e. voltage fluctuations in the frequency range 0-30 Hz./

A known structure of an arc furnace with a DC converter is schematically shown in Fig. 4. The primary winding of a regulated transformer 9, which feeds directly the converters, feeds at least one secondary winding 10, 11, each of which is connected to a rectifier bridge 12, usually in Graetz system, with the use of controlled semiconductor devices (thyristors). One of the output terminals of each of the rectifier units 12 is connected to the common moving electrode 7, and the other terminal is connected via an equalizing inductor 13 to the bottom electrode 14 directly in contact with the scrap 6.

The unit - rectifier bridge 12 in the Graetz circuit is shown in detail in Fig. 5. The rectifier unit 12 consists of two series of three thyristors 15, 16, mounted in a double parallelism or three-phase dual variation configuration, with a common cathode and a common anode, respectively, and cooperating with the other electrode via the same system (star or delta) of polyphase voltages. Each of the series of thyristors 15, 16 is switched on with the same common switch-on angle α or α2.

Generally, a plurality of six-pulse Graetz bridges are used to limit the harmonics introduced into the network, each of which is powered with a phase shift of the transformer secondary windings. Fig. 4 shows the two secondary windings 10, 11, shifted by 30 ° through a star-delta configuration, feeding the Graetz bridges, but this solution can be generalized to three Graetz bridges shifted by -20 °, 0 °, +20 °, or four Graetz bridges offset by 0 °, + 15 °, + 30 ° and 45 °. Each of the Graetz bridges is an elementary six-pulse transducer.

While the description contemplates a two-rectifier arrangement, for example two Graetz bridges, the invention also applies to more rectifiers fed from the shifted secondary windings.

Figures 6 and 7 correspond to figures 2 and 3 of the drawing but relate to powering a DC powered device.

As can be seen, the device operates at a constant current value with large fluctuations in the active power P, as opposed to an AC supply. A reduced, but still significant, fluctuation of the reactive power Q can be seen. Moreover, in average values, the reactive energy consumption is increased, which is usually dictated by the presence of a load regulator 17 for the lower transformer and a compensation unit 18. Such a DC device does not offer any significant improvements. to the AC device.

The reactive power AQ fluctuation in the various operating ranges is however smaller than in the case of AC power, as a result of which less voltage fluctuation and reduced flicker are introduced into the network. But in the case of insufficiently sized supply networks, those voltage fluctuations which are proportional to the reactive power consumed and

Inversely proportional to the short-circuit power of the network, they often remain too high, in particular because of flicker, and require very expensive units for additional correction.

If the instantaneous values of the reactive power consumption are taken into account, it can be seen that these values fluctuate slightly for direct current, since the current regulation performed by the converter is generally fast enough to limit the overcurrents at the moment of the fault to acceptable values.

The idea of the invention is based on combining the traditional structure of a thyristor converter with an energy flow control circuit 19, of the flywheel type, and enabling control corresponding to a substantially constant active power, the value of which will be adapted to the arc characteristics, to increase the efficiency of the furnace and at the same time to the capabilities of the converter.

The energy flow control circuit 19, of the flywheel type, is a circuit that blocks the flow of current in one direction and allows it to flow in the opposite, polarized manner that allows the stored electrical energy in the inductors while the transducer thyristors are conducted to flow into the furnace charge during the closure period, which follows the conduction period.

The energy flow control circuit used by the invention is of a type that allows the conduction time of the main thyristors to vary within the cycle as a function of the switch-on angle. The length of the on time period in this control circuit increases as the conduction time of the main thyristors decreases.

According to the invention, an energy flow control circuit 19, of the flywheel type, is used to increase the DC amplitude while reducing the DC voltage, and vice versa. The trigger angle of the thyristors increases, and the conduction time of the thyristor within the cycle decreases accordingly, and vice versa. Thus, an essentially constant active power is obtained without excessive development of the thyristors and the transformer.

In the illustrated example of an arc furnace, a flywheel type energy flow control circuit 19 is implemented by diodes 20 mounted between the midpoint of the transformer secondary winding 10 and each of the DC output terminals of the rectifier unit 12. The diode is of course biased in the opposite direction to the direction. conduction of the Graetz bridge diodes.

The use of diodes instead of thyristors in the energy flow control circuit 19 makes the transducer irreversible depending on the power. This property could be considered a disadvantage, but in the application under consideration, it is in fact a significant advantage.

As a result, in the classic Graetz bridge design, the transducer is a transducer that inverts two quarters of the current waveform (unidirectional direct current, but bidirectional direct voltage), which allows the current to be adjusted so that it is particularly efficient in the event of an arc fault, while limiting overcurrents to a reasonable amount . This is of course beneficial to the mains, but not to the short-circuit of the arc, which is to be removed as quickly as possible by melting the scrap due to the short-circuit. In a direct AC powered solution, the short-circuit overcurrent, in contrast, allows the scrap to melt more quickly, but then large reactive power problems arise.

Due to the fact that the transducer is power-irreversible, the solution according to the invention achieves numerous advantages. At the moment of an arc short circuit, the overcurrent caused by the irreversibility of the structure allows the short circuit scrap to melt faster without draining excess reactive power into the mains, as overcurrents will not be picked up by the flywheel type circuit diodes. Moreover, the above-mentioned overcurrents can be controlled and optimized depending on the properties of the electrodes in order to limit their wear.

According to another aspect of the invention, it is preferable to have a plurality of secondary windings in the transformer, such as windings 10 and 11, arranged offset, in particular parallel. The combination of the displacement effect and the operation of the flywheel-type circuit allows for a significant reduction in the reactive power consumed.

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More specifically, the known shifting technique, shown in the diagrams of Figures 9, 10, 11, consists of combining at least two transducers with the same structure and varying their activation angles in a way that affects the reactive power consumed. So there is talk of an offset or sequence control. This method is typically used in series connection on the DC side of two bridges as shown in Fig. 9. This is a series offset arrangement.

In the present case, the DC voltage supplying the arc is relatively small in relation to the capabilities of the high-power thyristors, and staggered mounting in series leads to a poor technological application of thyristors. Preference is given here to the offset parallel mounting type shown in Figures 10 and 11 in two possible variants.

In the parallel arrangement with the shift of Figs. 10 and 11, two Graetz bridges are connected, each of which is formed by two shifted half bridges, thus generating harmonic pairs, unlike the classic Graetz bridges, but with the intersection of the shifts of the internal half bridges in such a way to completely eliminate harmonic pairs at the output.

However, the classic parallel-offset circuit without a flywheel-type circumference, such as in Figs. 10 or 11, has the drawback of creating a risk of thyristor recomutation, in particular with a constant voltage close to zero. In order to reduce this risk, so far efforts have been made to limit the range of fluctuations of the switching-on angles, significantly reducing the reactive power gain.

The advantage of the combination of the displaced circuit with a flywheel-type circuit proposed by the invention is that the risk of recomutation is completely eliminated if the circuit of the flywheel type allows the conduction time of the main thyristors to be varied within the cycle. Thus, full accumulated gains of reactive power are achieved, with full operational safety.

Figure 6 is a complete schematic of a powered furnace with a diode zero point circle circuit and a parallel offset arrangement. This solution aims, through the possibility of appropriate control, to minimize the reactive power consumed and thus the reactive energy at any time (ie for a given operating point). This case corresponds to the situation in which the priority parameter for optimization is the lowest possible average value of reactive power consumed.

As a result, acts on the control parameters, which are the angles α and switching αι ₂ in such a way as to obtain the maximum displacement, ie. The maximum value (αι -). Other regulation; because it can be manipulated separately to the two angles α and switching αι _2, it is possible to control two functions, which adjusts the current or active power during operation.

Figures 12 and 13 show the performance characteristics obtained with the scheme of Figure 8, in the case of the first variant, with the minimized average reactive power consumed.

Figure 12 shows a voltage / current characteristic which is similar to that for direct AC supply, i.e. operation is at constant power. The dashed zone represents an improvement in properties compared to the classic transducer.

Figure 13 shows the reactive power / active power characteristic for a given operating point, i.e. for a constant current I, or for a rectified current Id that does not change and a variable equivalent arc resistance. This characteristic shows that the reactive power absorbed is more than three times lower than the maximum active power, which results in a significant improvement over the previous situation for AC or DC supply (compare with Figures 3 and 7). At the bottom of Fig. 13 are further provided angle graph of the variability of integration αι and α ₂ depending on the supplied power.

The advantages obtained with the invention with such a scheme are very significant. Removal of the converter transformer regulator, significant reduction of the compensating and filtering battery power (by about two times), significant increase in efficiency in terms of the number and duration, reduction of voltage fluctuations introduced into the mains, reduction of flicker, reduction of converter losses, reduction of harmonics and disturbances high-frequency signals produced by the transducer.

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The only condition for obtaining these improvements is the use of a flywheel-type diode energy control circuit 19 and the selection of appropriate dimensions for the connections, i.e. equalizing inductances, of the electrode leads downstream of the transducer, so that their size matches the permissible overcurrents. In practice, this leads to a reduction in the investment cost of the electric arc furnace.

In the second variant, the ability to control the control parameters, i.e. the activation angles α1 and α2, was selected, depending on the supplied power, in such a way as to focus no longer on the reduction of the average reactive power consumed, but on the reduction of the reactive power fluctuations around its average value , and therefore a reduction in flicker, allowing the consumption of a slightly higher average reactive power. A converter capable of controlling the active power is then used, while at the same time consuming a constant reactive power which can be compensated by the simple compensation unit 18 of Fig. 4 with fixed capacitances.

In the second variant, the basic diagram of Fig. 8 (i.e. the use of a Graetz bridge and an energy flow control circuit 19, of the offset flywheel type) with the possibility of controlling the activation angles αι and α2, but unlike the previous case, does not seek the maximum value shifts of the activation angles α2 and α1.

Figure 14 shows the characteristic reactive power Q / active power P for a given operating point, i.e. for a constant direct current I or a rectified current Ii and a variable equivalent arc resistance. In Fig. 14, characteristic I corresponds to a classical diagram of a transducer without an energy flow control circuit 19, flywheel type, and without offset, i.e. the characteristic of Fig. 7. Characteristic II corresponds to a diagram with an energy flow control circuit 19, flywheel type, but without no shift, i.e. the diagram of Fig. 8 with the condition of equal turn-on angles (αι = α2). The characteristic III corresponds to the same scheme, at the maximum shift, ie for the maximum value of the difference of the switching on angles (α11 - α2) (characteristic shown in Fig. 13).

An infinitely many different characteristics can be obtained between the zero offset and the maximum offset, all of which lie within the hatched zone between characteristics II and III.

More specifically, these characteristics are determined by the following equations:

P = R x Ii = EixIi = Ei xIi xfP (α1, α2)

Q = Eio x Ii x fq (α1, α2)

P active power

Q reactive power

R is the equivalent arc resistance

Ii rectified current

Ei rectified operating voltage

Eio rectified open-circuit voltage for zero shift fp, fq functions of switch-on angles

Thus, for a set duty point, i.e., for R and Ii data, P will be determined, but Q will be controlled to some extent.

For example, you can force a constant reactive power Q, which corresponds to the IV curve, or also all the characteristics included in the shaded area. Then Id, P and Q are computed and α1 - α2 are determined directly.

Moreover, if, for example, as indicated above, an attempt is made to maintain a substantially constant active power causing an increase in Ii with a decrease in the equivalent arc resistance (or vice versa), the operating point will shift and the characteristic curve will shift in the diagram.

Let us assume that by solving a system of two equations with two unknowns, we will ensure, for the operating point for a given current, a characteristic corresponding to that in Fig. 5 marked with V, i.e., the function Q = f (P) increases monotonically. In this case, while the current Ii varies, the characteristic V moves as illustrated in Fig. 15, delimiting the family

174,299 characteristics corresponding to each set operating point (Id current). One can therefore choose the appropriate range of changes Id = f (R) decreasing monotonically, that Q will remain constant despite the fluctuations in Id.

In the latter case, therefore, a converter is used that offers, for all operating points (or at least for a large majority of them), the simultaneous substantial constancy of the reactive power Q and the active power P.

In order to realize the dependencies defined by the aforementioned system of equations, the rectifier units 12 are equipped with controlled components for determining the size of the control parameters constituting the activation angles α and C2 of the regulated semiconductor devices T1, T2, to minimize the reactive power consumed at the minimum average value per cycle of the set operating point, or to minimize fluctuations of reactive power around its mean value at the set value of the rectified current Id, possibly to maintain the average value of the reactive power consumed at a constant level in the event of fluctuations in the rectified current Id.

Moreover, numerous improvements can be applied to the properly illustrated schematic diagram. Firstly, cyclic changes in the direction of the shifting of two bridges (αι + α2 for one bridge and α2 + α for the other) can be advantageously planned in order to equalize the heating of the semiconductors and reduce the risk of the appearance of harmful DC components for the magnetic induction of the transformer.

The period of these cyclic changes can be calculated depending on the thermal time constant of the semiconductors and the constant component allowed for the transformer. As for the moment of change within the cycle, it should be chosen so as to minimize the transient amplitude of the commutations arising, either on the line side or on the DC side. These transitions should not be greater than the natural fluctuations of the arc.

Moreover, it is also possible to design double current regulation by realizing separate regulation of the currents of each Graetz bridge, in parallel, or regulation of the sum of the two currents and their equality simultaneously (crossover regulation).

In case of imbalance between two bridges, bad compensation of harmonic pairs may occur and it may be necessary to limit the imbalance (limiting αι-α1 and α2-α2 ') so as to limit the harmonic pairs to an acceptable level. Such a risk may in particular occur when the two Graetz bridges are not connected exactly in parallel, namely in the case where the connection is made not at the level of the lower equalizing inductance terminals, but at the level of two bottom electrodes electrically connected through the base of the tank.

In addition, multiple components of the same type, such as that shown in Fig. 9, can be interconnected by feeding them through phase-shifted transformers to reduce harmonics. The phase shift of the transformers is advantageously at the level of the primary windings, for example with a zigzag triangle or a triangle with a center socket winding.

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FIG. 13

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FIG. 8

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in FIG. 6

ABOUT-

ΔΙ = Ο

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FIG1

FIG. 2

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Claims (10)

Patent claims 1.Power converter for the direct current supply of an electric arc furnace, provided with at least one transformer fed from the primary winding side with a three-phase alternating current, which has at least one secondary winding through which three-phase current flows and which is connected to a rectifier unit whose direct current the output is connected to the furnace charge, the rectifier unit associated with each transformer secondary winding being equipped with controllable semiconductor devices characterized in that between the midpoint of the transformer secondary winding (10, 11) and the DC output terminals of the rectifier unit (12) is the energy flow control circuit (19) of the flywheel type is turned on, and the controlled semiconductor devices (T1, T2) are turned on with different switching on angles (al, a2) to increase and decrease the conduction time of the semiconductor devices (T1, T2) in the on state. 2. The transducer according to claim The method of claim 1, wherein the energy flow control circuit (19) is a diode circuit (20), the transducer being irreversible. 3. The transducer according to claim The method of claim 1, characterized in that the rectifier units (12) connected to the secondary windings (10, 11) of the transformer are connected with each other in a system with an offset, with different switching on angles (α1, α2) of the controlled semiconductor devices (T1, T2) . 4. The transducer according to claim A device according to claim 3, characterized in that the rectifier unit (12) is provided with a subassembly for determining the switching-on angles (α1, α2) of the controlled semiconductor devices (T1, T2) adapted to minimize the reactive power consumption with a minimum average value per cycle for a given operating point. 5. The transducer according to claim 1 3. A method according to claim 3, characterized in that the rectifier unit (12) is provided with a subassembly for determining the switching-on angles (α1, α2) of the controlled semiconductor devices (T1, T2) adapted to minimize the fluctuations of the reactive power around its average value (Q) for the given rectified current ( Id). 6. The transducer according to claim 3. A method according to claim 3, characterized in that the rectifier unit (12) is provided with a sub-assembly for determining the switching-on angles (α1, α2) of the controlled semiconductor devices (T1, T2) adapted to keep the average reactive power consumption (Q) constant in the event of fluctuations in the rectified current (Id). 7. The transducer according to claim 3. The process of claim 3, characterized in that the rectifier units (12) are connected to each other in a parallel offset arrangement. 8. The transducer according to claim 1 3. The method of claim 3, characterized in that the rectifier units (12) are adapted to cyclically change the direction of displacement. 9. The transducer according to claim 1 3. A method according to claim 3, characterized in that it is additionally provided with current regulation units of two rectifier units (12), the magnitudes of these currents being substantially equal. 10. The transducer according to claim 1 A method according to claim 1, characterized in that it is provided with a set of rectifier devices (12), the respective transformers of which are energized with a phase shift.